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偏微分方程地構建科學、工程學和其他領域的數學模型的主要手段。一般情況下，這些模型都需要用數值方法去求解。本書提供瞭標準數值技術的簡明介紹。藉助拋物綫型、雙麯綫型和橢圓型方程的一些簡單例子介紹瞭常用的有限差分方法、有限元方法、有限體方法、修正方程分析、辛積分格式、對流擴散問題、多重網絡、共軛梯度法。利用極大值原理、能量法和離散傅裏葉分析清晰嚴格地處理瞭穩定性問題。本書全麵討論瞭這些方法的性質，並附有典型的圖像結果，提供瞭不同難度的例子和練習。

本書可作為數學、工程學及計算機科學專業本科教材，也可供工程技術人員和應用工作者參考。

K.W.Morton，牛津大學退休教授，曾任教於數值分析學術重鎮牛津大學計算實驗室。現為巴斯大學兼職教授。主要研究領域為有限差分、有限元和有限體方法。Morton有著豐富的教學經驗，他存數值分析領域的理論研究和實際應用方麵的成就也廣為人知。他曾擔任數值分析界最高榮譽Leslie Fox奬評委會主席。

D.F.Mayers，曾任職於牛津大學計算實驗室，是已故數值分析先驅Leslie Fox的長期閤作者。除本書之外，他還著有廣泛采用的教材An Introduetio to Numerical Analysis。

1 Introduction2 Parabolic equations in one space variable 2.1 Introduction 2.2 A model problem 2.3 Series approximation 2.4 An explicit scheme for the model problem 2.5 Difference notation and truncation error 2.6 Convergence of the explicit scheme 2.7 Fourier analysis of the error 2.8 An implicit method 2.9 The Thomas algorithm 2.10 The weighted average or 0.method 2.11 A maximum principle and convergence 2.12 A three.time.level scheme 2.13 More general boundary conditions 2.14 Heat conservation properties 2.15 More general linear problems 2.16 Polar co.ordinates 2.17 Nonlinear problems Bibliographic notes Exercises3 2.D and 3.D parabolic equations 3.1 The explicit method in a rectilinear box 3.2 An ADI method in two dimensions 3.3 ADI and LOD methods in three dimensions 3.4 Curved boundaries 3.5 Application to general parabolic problems Bibliographic notes Exercises4 Hyperbolic equations in one space dimension 4.1 Characteristics 4.2 The CFL condition 4.3 Error analysis of the upwind scheme 4.4 Fourier analysis of the upwind scheme 4.5 The Lax.Wendroff scheme 4.6 The Lax.Wendroff method for conservation laws 4.7 Finite volume schemes 4.8 The box scheme 4.9 The leap.frog scheme 4.10 Hamiltonian systems and symplectic integration schemes 4.11 Comparison of phase and amplitude errors 4.12 Boundary conditions and conservation properties 4.13 Extensions to more space dimensions Bibliographic notes Exercises5 Consistency, convergence and stability 5.1 Definition of the problems considered 5.2 The finite difference mesh and norms 5.3 Finite difference approximations 5.4 Consistency, order of accuracy and convergence 5.5 Stability and the Lax Equivalence Theorem 5.6 Calculating stability conditions 5.7 Practical (strict or strong) stability 5.8 Modified equation analysis 5.9 Conservation laws and the energy method of analysis 5.10 Summary of the theory Bibliographic notes Exercises 6 Linear second order elliptic equations in two dimensions 6.1 A model problem 6.2 Error analysis of the model problem 6.3 The general diffusion equation 6.4 Boundary conditions on a curved boundary 6.5 Error analysis using a maximum principle 6.6 Asymptotic error estimates 6.7 Variational formulation and the finite element method 6.8 Convection.diffusion problems 6.9 An example Bibliographic notes Exercises7 Iterative solution of linear algebraic equations 7.1 Basic iterative schemes in explicit form 7.2 Matrix form of iteration methods and their convergence 7.3 Fourier analysis of convergence 7.4 Application to an example 7.5 Extensions and related iterative methods 7.6 The multigrid method 7.7 The conjugate gradient method 7.8 A numerical example: comparisons Bibliographic notes ExercisesReferencesIndex
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